The Finite-dimensional Modular Lie Superalgebra Ω,г,(?)

The present thesis is devoted to studying modular Lie superalgebras. The study ofLie superalgebras mainly contains structures, classifications and representations. For thedi?erent characteristic number, Lie superalgebras can be divided into non-modular Liesuperalgebras (i.e., Lie superalgebras over fields of characteristic zero) and modular Liesuperalgebras (i.e., Lie superalgebras over fields of prime characteristic). The theory ofnon-modular Lie superalgebras has experienced a rather vigorous development through-out the combined efforts of numerous mathematicians and Physicists. The first majorachievements were the classi?cations completed by V. G. Kac. But the investigation ofmodular Lie superalgebras just begin in recent years. The results pertaining to modularLie superalgebras are not so plentiful. In particular, the classi?cation of simple modularLie superalgebras has not obtained signi?cant advances at present stage. So constructingnew ?nite-dimensional simple modular Lie superalgebras is of great importance.In Chapter 1, we review brie?y the background information and progress on modularLie superalgebras. Then the basic concepts and certain known results on modular Liesuperalgebras which will be used in this paper are reviewed.Prof. Zhang Yongzheng constructed a class of ?nite-dimensional simple modular LiesuperalgebrasΩand determined its derivation superalgebra in 2009. Moreover, it wasproved thatΩwas not isomorphic to the known modular Lie superalgebras of Cartantype. The purpose of Chapter 2 is to continue the investigation of ?ltration structure ofmodular Lie superalgebra ?. We proved that the ?ltration ofΩis invariant under theautomorphism group by the method of minimal dimension of image spaces. Furthermore,we obtain that the integers in the de?nition of modular Lie superalgebraΩare intrinsicand therefore, classify the modular Lie superalgebra ofΩtype in the sense of isomorphism.We discuss the associative form and restrictiveness of modular Lie superalgebrasΩwhichis motivated by the methods in the modular Lie superalgebra of Cartan type situation. It is proved that ? has no nonsingular Killing form.Based on the corresponding underlying superalgebra of ?, two families of ?nite-dimensional modular Lie superalgebrasΓand D are constructed in chapter 3 and 4,respectively, and their simplicity and generators are studied. By giving the generatorsset we determine the homogeneous superderivations of modular Lie superalgebrasΓandD. Furthermore, we determine completely the derivation superalgebra ofΓand D. Itis proved that they are not isomorphic to any known modular Lie superalgebras of Car-tan type. Moreover, we determine the Lie superalgebraΓthat possesses a nonsingularassociative form, D has no nonsingular associative form and neitherΓnor D has nonsin-gular Killing form. We obtain the su?cient and necessary conditions thatΓand D arerestricted Lie superalgebras.